Cut each chicken breast in half width-wise so you have 4 thin chicken cutlets. In a small bowl, combine zest, black pepper, and salt. Rub evenly all over chicken cutlets.
Place 1 tablespoon butter and olive oil in a large 9-inch skillet and heat over medium-high heat. Add chicken to pan and sear 3-4 minutes on each side until chicken registers 165F. Transfer to a plate and lightly brush lemon juice over cutlets. Keep warm and set aside.
Wipe pan clean. Add remaining 2 tablespoons butter to same pan and melt over medium heat. Add garlic and saute 1-2 minutes until fragrant. Whisk in flour and cook several minutes until golden brown.
Slowly begin drizzling in milk and whipping cream, whisking constantly until sauce is smooth. Cook sauce over medium heat, stirring occasionally until sauce is bubbly and thickened. Remove from heat and whisk in Parmesan cheese until melted.
Fold linguine into sauce until evenly coated, thinning sauce with pasta water if needed. Season with salt and pepper to taste if needed. Slice chicken cutlets into thin slices and arrange on top of linguine.
Top skillet with parsley, additional lemon zest, and Parmesan. Serve warm and enjoy!